11Nash Equilibrium - Existence I - Nash Equilibrium:...

Nash Equilibrium: Existence Let G = ( N ; S 1 ,...,S n ; u 1 ,...,u n ) be a strategic form game with N = { 1 ,...,n } . Theorem G has a Nash Equilibrium if, for every i ∈ N , (a) S i is a nonempty, compact, convex subset of R m for some integer m ; (b) u i is quasiconcave on S i for every s-i ∈ S-i and is continuous on S . Suppose that G is a ﬁnite game (each S i is ﬁnite). I denote its mixed extension by Δ G = (Δ S 1 ,..., Δ S n ; u 1 ,...,u n ), where for each i ∈ N , Δ S i is the set of probability distributions over S i , and u i is a vN-M utility. I denote a typical element of Δ S i by σ i ; the probability that player i plays ’pure’ strategy b s i ∈ S i by σ i ( b s i ); a mixed strategy proﬁle by σ = ( σ 1 ,...,σ n ), where σ i ∈ Δ S i for all i ∈ N ; and the set of feasible mixed strategy proﬁles for Δ G by Δ S . Note that the expected utility for player i when proﬁle σ is played is given by U i ( σ ) = X s ∈ S u i ( s ) σ 1 ( s 1 ) σ

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